1. **Problem Statement:** We have a sample of times (in seconds) taken by employees to complete a task: 63, 229, 165, 77, 49, 74, 67, 59, 66, 102, 81, 72, 59, 74, 61, 82, 48, 70, 86. 2. **Stem-and-Leaf Diagram:** - Stem represents the tens digit, leaf the units digit. - Sorted data: 48, 49, 59, 59, 61, 63, 66, 67, 70, 72, 74, 74, 77, 81, 82, 86, 102, 165, 229. - Stem & leaf: 4 | 8 9 5 | 9 9 6 | 1 3 6 7 7 | 0 2 4 4 7 8 | 1 2 6 10 | 2 16 | 5 22 | 9 This shows most data clustered between 40 and 90 seconds, with some high outliers. 3. **Arithmetic Mean:** $$\text{Mean} = \frac{\sum x_i}{n} = \frac{63 + 229 + 165 + 77 + 49 + 74 + 67 + 59 + 66 + 102 + 81 + 72 + 59 + 74 + 61 + 82 + 48 + 70 + 86}{19}$$ Calculate sum: $$63 + 229 + 165 + 77 + 49 + 74 + 67 + 59 + 66 + 102 + 81 + 72 + 59 + 74 + 61 + 82 + 48 + 70 + 86 = 1635$$ So, $$\text{Mean} = \frac{1635}{19} \approx 86.05$$ 4. **Mode:** The most frequent values are 59 and 74 (each appears twice), so the data is bimodal with modes 59 and 74. 5. **Median:** Sorted data has 19 values, median is the 10th value: 10th value = 72 So, median = 72. 6. **Quartiles:** - Q1 (25th percentile): median of first 9 values: 48, 49, 59, 59, 61, 63, 66, 67, 70 Median of these 9 values is 61 (5th value). - Q3 (75th percentile): median of last 9 values: 74, 74, 77, 81, 82, 86, 102, 165, 229 Median is 82 (5th value). 7. **80th Percentile:** Position = $0.8 \times (19 + 1) = 16$th value in sorted data. 16th value = 82 Interpretation: 80% of employees took 82 seconds or less to complete the task. 8. **Variance and Standard Deviation:** Variance formula: $$s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$$ Calculate squared deviations and sum: Sum of squared deviations $\approx 29588.95$ Variance: $$s^2 = \frac{29588.95}{18} \approx 1643.83$$ Standard deviation: $$s = \sqrt{1643.83} \approx 40.54$$ 9. **Coefficient of Variation (CV):** $$CV = \frac{s}{\bar{x}} = \frac{40.54}{86.05} \approx 0.471$$ This measures relative variability; a CV of 0.471 means the standard deviation is about 47.1% of the mean, indicating moderate variability. 10. **Box-Plot Construction:** - Minimum: 48 - Q1: 61 - Median: 72 - Q3: 82 - Maximum: 229 - Outliers: Values beyond $Q3 + 1.5 \times IQR$ or below $Q1 - 1.5 \times IQR$ where $IQR = Q3 - Q1 = 21$ Upper fence: $82 + 1.5 \times 21 = 113.5$ Lower fence: $61 - 1.5 \times 21 = 29.5$ Values above 113.5 are outliers: 165, 229 11. **Skewness and Outliers:** - Data is right-skewed due to high outliers (165, 229). - Most data is concentrated on the lower end with a long tail to the right. Final answers: - Mean $\approx 86.05$ - Mode: 59 and 74 - Median: 72 - Q1: 61, Q3: 82 - 80th percentile: 82 - Variance $\approx 1643.83$ - Standard deviation $\approx 40.54$ - Coefficient of variation $\approx 0.471$ - Outliers: 165, 229 - Data is right-skewed.